The calendar for the year 1993 will be same for the year ______.
2022
The calendar for the year 1993 will be same for the year ______.
- A.
2004
- B.
2010
- C.
1999
- D.
2021
Attempted by 1 students.
Show answer & explanation
Correct answer: C
Concept
A calendar of one year repeats in a later year only when BOTH conditions hold: (1) the two years have the same leap-year status (both ordinary, or both leap), and (2) the total number of “odd days” over the years that elapse between them is a multiple of 7, so that 1 January falls on the same weekday. An odd day is the remainder when a year’s day-count is divided by 7: an ordinary year (365 days) has 1 odd day; a leap year (366 days) has 2 odd days.
Application to 1993
1993 is an ordinary year (not divisible by 4) whose 1 January is a Friday. To move 1 January forward to the same weekday, add the odd days of the elapsed years 1993, 1994, … until the running total is a multiple of 7:
1993 (ordinary) → +1, running total = 1
1994 (ordinary) → +1, running total = 2
1995 (ordinary) → +1, running total = 3
1996 (leap) → +2, running total = 5
1997 (ordinary) → +1, running total = 6
1998 (ordinary) → +1, running total = 7 ≡ 0 (mod 7)
After the 6 years 1993–1998 the elapsed odd days total 7 ≡ 0 (mod 7), so 1 January 1999 is again a Friday; and 1999 is itself an ordinary year, matching the leap status of 1993. Both conditions hold, so the calendar of 1999 is identical to that of 1993 — and 1999 is the first such year, the year asked for.
Cross-check (the other offered years)
2004: this is a leap year while 1993 is ordinary, and 1 January 2004 is a Thursday rather than a Friday — both the leap-status and the same-weekday conditions fail, so 2004 is ruled out.
2010 and 2021: these are also ordinary years that begin on a Friday, so each shares the 1993 calendar; they are later points in the same repeat cycle rather than the first return.
Result: 1999 — the first year whose calendar is identical to 1993.
Note: 1993, 1999, 2010 and 2021 all share an identical calendar, so three of the four offered years are technically valid same-calendar years; by the standard convention the single intended answer is the first repeat, 1999.